3 The Schrödinger equation

If the electron in an atom of hydrogen is a standing wave, as de Broglie had assumed, why should it be confined to a circle? After the insight that particles can behave like waves, which came ten years after Bohr’s quantization postulate, it took less than three years for the full-fledged (albeit still non-relativistic) quantum theory to be formulated, not once but twice in different mathematical attire, by Werner Heisenberg in 1925 and by Erwin Schrödinger in 1926.

Let’s take a look at where the Schrödinger equation, the centerpiece of non-relativistic quantum mechanics, comes from. Figure 2.3.1 illustrates the properties of a traveling wave ψ of Amplitude A and phase φ = kx − t. The wavenumber k is defined as 2π/λ; the angular frequency ω is given by 2π/T. Hence we can also write

φ = 2π [(xλ) − (t/T)].

Keeping t constant, we see that a full cycle (2π, corresponding to 360°) is completed if x increases from 0 to the wavelength λ. Keeping x constant, we see that a full cycle is completed if t increases from 0 to the period T. (The reason why 2π corresponds to 360° is that it is the circumference of a circle or unit radius.)

Figure 2.3.1 The slanted lines represent the alternating crests and troughs of ψ. The passing of time is indicated by the upward-moving dotted line, which represents the temporal present. It is readily seen that the crests and troughs move toward the right. By focusing on a fixed time, one can see that a cycle (crest to crest, say) completes after a distance λ. By focusing on a fixed place, one can see that a cycle completes after a time T.

The mathematically simplest and most elegant way to describe ψ is to write

ψ = [A:φ] = [A:kx − ωt].

This is a complex number of magnitude A and phase φ. It is also a function ψ(x,t) of one spatial dimension (x) and time t.

We now introduce the operators ∂x and ∂t. While a function is a machine that accepts a number (or several numbers) and returns a (generally different) number (or set of numbers), an operator is a machine that accepts a function and returns a (generally different) function. All we need to know about these operators at this point is that if we insert ψ into ∂x, out pops ikψ, and if we insert ψ into ∂t, out pops −iωψ:

∂xψ = ikψ, ∂tψ = −iωψ.

If we feed ikψ back into ∂x, out pops (not unexpectedly) (ik)2ψ = −k2ψ. Thus

(∂x)2ψ = −k2ψ.

Using Planck’s relation E = ℏω and de Broglie’s relation p = h/λ = ℏk to replace ω and k by E and p, we obtain

∂tψ = −i(E/ℏ)ψ, ∂xψ = i(p/ℏ)ψ, (∂x)2ψ = −(p/ℏ)2ψ,

or

(2.3.1) Eψ = iℏ ∂tψ, pψ = (ℏ/i)∂xψ, p2ψ = −ℏ2(∂x)2ψ.

We now invoke the classical, non-relativistic relation between the energy E and the momentum p of a freely moving particle,

(2.3.2) E = p2/2m,

where m is the particle’s mass. We shall discover the origin of this relation when taking on the relativistic theory. The right-hand side is the particle’s kinetic energy.

Multiplying Eq. (2.3.2) by ψ and using Eqs. (2.3.1), we get

(2.3.3) iℏ ∂tψ = −(ℏ2/2m) (∂x)2ψ.

This is the Schrödinger equation for a freely moving particle with one degree of freedom — a particle capable of moving freely up and down the x-axis. We shouldn’t be surprised to find that Eq. (2.3.3) imposes the following constraint on ψ:

(2.3.4) ω = ℏk2/2m.

This is nothing else than Eq. (2.3.2) with E and p replaced by ω and k according to the relations of Planck and de Broglie.

We have started with a specific wave function ψ. What does the general solution of Eq. (2.3.3) look like? The question is readily answered by taking the following into account: If ψ1 and ψ2 are solutions of Eq. (2.3.3), then for any pair of complex numbers a,b the function ψ = aψ1 + bψ2 is another solution. The general solution, accordingly, is

(2.3.5) ψ(x,t) = (1/√(2π)) ∫dk [a(k):kx − ω(k)t].

The factor (1/√(2π)) ensures that the probabilities calculated with the help of ψ are normalized (that is, the probabilities of all possible outcomes of any given measurement add up to 1). The symbol ∫dk indicates a summation over all values of k from k=−∞ to k=+∞: every value contributes a complex number a(k)[1:kx − ω(k)t], where ω(k) is given by Eq. (2.3.4).

If the particle is moving under the influence of a potential V, the potential energy qV (q being the particle’s charge) needs to be added to the kinetic energy (the right-hand side of Eq. 2.3.2). The Schrödinger equation then takes the form